In channels with steep slopes, the flow is supercritical and the formation of roll waves is expected. In order to evaluate the flow behaviors in steep open channels, the Azad dam spillway is simulated numerically and its flow characteristics are compared with the physical model tests. In this study, the hydraulic variables including hydraulic depth and water surface profile are studied along the steep open channels. The k-*ɛ* turbulence model is considered in this study. It is shown that in the conditions in which the angle of the wave forehead is less than 35 degrees, the formation of roll waves is inevitable.

## INTRODUCTION

Usually the regime of flow in the steep open channels is supercritical. In supercritical flows, the instability on the free surface is intensified, which is the prerequisite for roll wave formation. Roll waves in an open channel are formed in which the gravity forces are dominant. This phenomenon might be studied by numerical and physical models. Nowadays, numerical models are used more frequently than experimental models as they allow the spending of less time and money.

*et al.*1995). In the studies by Gualtieri & Chanson (2007) the distributions of the void fraction and the air bubbles count rate were recorded in the hydraulic jump for the Froude numbers varying from 5.2 to 14.3, which is consistent with previous studies. Also, the data demonstrated that the upper boundary of the air diffusion layer tends systematically to linearly increase by increasing the distance from the jump toe for a fixed inflow Froude number expressed as follows: In the above equation,

*Fr*represents Froude number,

*g*gravitational acceleration,

*V*and

*y*are the velocity and depth of flow, respectively.

A comprehensive study for simulating flow over a stepped spillway, especially upstream of the inception point of air entrainment was achieved by Bombardelli *et al.* (2011) and Chakib (2013), using FLUENT software and a volume of fluid model. They found a good agreement between the experimental data and the numerical results for velocity vectors, streamlines and total pressures over the spillway. Bazargan & Aghebatie (2015) used the standard k-*ɛ* model to simulate turbulence and volume of fluid model to track the free surface of the chute for different discharges in a three-dimensional numerical model. Velocities and water surface profiles obtained from a numerical model are compared with the physical model data. The results showed that the numerical results and the measured data are basically consistent and can provide a reliable basis for chute optimization design. Also, they showed that for rectangular chutes with the ratio of hydraulic depth over the chute width interior of 0.16, the flow becomes unstable and roll waves are formed. Bayon-Barrachina & López-Jiménez (2015) studied the hydraulic jump in a horizontal rectangular canal by using OpenFOAM software and different turbulence models such as standard k − ɛ, RNG k − ɛ, and SST k −ω turbulence models. Accuracies of about 98% are achieved in the numerical model for different parameters such as sequent depths, efficiency, roller length, free surface profile, etc. in comparison with the physical model tests. In the present study, Azad Dam spillway is simulated numerically and compared with the prototype results. Design essentials and criteria for producing the roll waves are investigated. In this study, the previous criteria are evaluated and further criteria are proposed using the FLUENT software.

## METHODS

### Data provided for numerical simulation

^{3}/s. The spillway consisted of an ogee of 9.78 m length at 1,465 m above sea level. The ogee curve is defined by , so the beginning level of the chute was located 1,458.13 m above sea level. The chute is 30 m in width and has two different slopes; 5% and 36.4% with horizontal lengths equal to 45.4 and 198.31 m in each part, respectively. The chute ends in a flip bucket with a radius of 15 m and a horizontal length of 10.58 m. A schematic view of the spillway is shown in Figure 1.

### Simulation of free surface flow

*μ*parameters, the momentum equation is a function of the volumetric ratios of all phases (FLUENT 2004; Houichi

*et al.*2006; Yeoh & Tu 2010). in which, is the fluid density,

*μ,*, and are the dynamic viscosity, fluid pressure, velocity vector and finally the gravitational body force, respectively. Also, in the above equations, is the transpose sign.

*y*and

^{+}*U**are non-dimensional numbers presented in Equations (3) and (4), which determine whether the influences in the wall adjacent cells are laminar or turbulent, hence indicating the part of the turbulent boundary layer that they resolve (Akoz & Kirkgoz 2009). In the above equations, is the average velocity,

*y*is the distance from the wall and is the wall stress. The logarithmic velocity law for is valid, which in FLUENT software separates this boundary, and the following equation governs this layer.

### Turbulence model

One of the main characteristics of turbulent flow is the fluctuating velocity fields. These fluctuations cause mixing of transported quantities like momentum, energy and species concentration and thereby also fluctuations in the transported quantities. Because of the small scales and high frequencies of the fluctuations, they are too computationally expensive to simulate directly in practical engineering situations. Instead, the instantaneous governing equations are time-averaged to remove the small scales and the result is a set of less expensive equations containing additional unknown variables. These unknown (turbulence) variables are determined in terms of modeled variables in turbulence models.

*k*-

*ɛ*model. It is one of the two-equation models which are considered the simplest of the so called complete models of turbulence. Ever since it was proposed in 1974, its popularity in industrial flow simulations has been explained by its robustness, economy and reasonable accuracy for a wide range of turbulent flows. The model is a semi-empirical model based on modeled transport equations for the turbulence kinetic energy (

*k*) and its dissipation rate (

*ɛ*). Applied equations in the model are as follows (Kherbache

*et al.*2013; Kositgittiwong

*et al.*2013): in which

*μ*indicates the dynamic viscosity, is the fluid density, is the eddy viscosity, is an empirical coefficient equal to 0.09. The coefficients of the above equations are . is a constant considered 1 for the flow component parallel to gravity direction and equals zero for the flow components perpendicular to the gravity direction. Terms and are the turbulent Prandtl numbers for and , respectively.

In the above equations, *b*, *T* and are the coefficient of thermal expansion, the temperature and the eddy viscosity, respectively. Term is the turbulent Prandtl number for energy and the variables and are velocities in the and directions, finally and correspond to *x* and *y* coordinates.

### Verification of numerical model

^{3}/s over the spillway, chute and flip bucket of Azad Dam based on numerical modeling and experimental simulation results. As shown in these figures, a good agreement is achieved between the numerical and the hydraulic model tests. The mean relative errors of water surface profiles between simulated and experimental values for standard

*k*-

*ɛ*, RNG

*k*-

*ɛ*, and realizable

*k*-

*ɛ*turbulence models are about 6.2%–9.33% and 11.4%, respectively. The turbulence model that is used in the present study is the standard

*k*-

*ɛ*model.

### Further simulations

After accomplishing the verification tests, the study is focused on the different slopes and shapes. The flow field was modeled in Gambit using unequal hexahedron rectangular elements. Then, the coordinates of the created nodes (not the created rectangular mesh grid) were exported from Gambit into FLUENT. Steep open channels modeled in this study are considered as three-dimensional channels that discharge 15 to 30 m^{3}/s and the longitudinal slope varying from 17 to 20%. Inlet boundary condition was defined as mass-flow-inlet and as zero pressure for the outlet and upper space of the models. The number of elements varied from about 172,238 for the channel with the diameter of 4.5 m and a slope of 17% and a flow discharge of 30 m^{3}/s, up to about 137,745 for the channel with the diameter of 3.6 m and a slope of 20% and input discharge of 15 m^{3}/s. The logarithmic distribution of velocity is valid only for the ranging from 30 to 300 according to Kawai & Larsson (2012). The mesh sizes are organized to produce situated in the valuable ranges.

*M*

_{ɛ}=

*M*

_{L}

^{0.166}in which

*M*

_{ɛ}and

*M*

_{L}are the roughness scale and length scale, respectively) was considered for the prototype. Time steps varied from 0.0001 up to 0.01 s from the beginning of simulation to the end of the simulation. This value obtained is about 210. A schematic view of the grid mesh with the optimum mesh size is shown in Figure 4.

Grid dimensions are considered from 0.07 to 0.15 m in the real model. In the numerical model, the steady state is evaluated by analyzing the vertical distribution of the velocity at the given points in which the input and output discharges must be equivalent. A maximum difference of 0.05% between input and output discharges was considered for the steady state conditions.

## RESULTS AND DISCUSSION

^{3}/s.

After the validation and calibration tests of the numerical model, using experimental results including flow depth and velocity and wave formation, the numerical model is used to evaluate the hydraulic properties of the flow and the criteria for the formation of rolling waves in different conditions, which are presented in Table 1.

. | . | Point length (m) . | ||||||
---|---|---|---|---|---|---|---|---|

Elliptical channel diameter (c)
. | Flow depth (D)
. | 19 . | 20 . | 25 . | 30 . | 35 . | 40 . | 45 . |

0.799 | 0.748 | 0.600 | 0.524 | 0.476 | 0.441 | 0.415 | ||

| 0.177 | 0.166 | 0.133 | 0.116 | 0.106 | 0.098 | 0.092 | |

0.811 | 0.758 | 0.607 | 0.530 | 0.478 | 0.447 | 0.428 | ||

| 0.193 | 0.181 | 0.144 | 0.126 | 0.113 | 0.106 | 0.102 | |

0.856 | 0.802 | 0.643 | 0.561 | 0.511 | 0.473 | 0.445 | ||

| 0.204 | 0.191 | 0.153 | 0.133 | 0.121 | 0.112 | 0.106 | |

0.851 | 0.796 | 0.637 | 0.556 | 0.502 | 0.469 | 0.449 | ||

| 0.212 | 0.199 | 0.159 | 0.139 | 0.125 | 0.117 | 0.112 | |

0.945 | 0.885 | 0.707 | 0.618 | 0.558 | 0.522 | 0.499 | ||

| 0.262 | 0.245 | 0.196 | 0.171 | 0.155 | 0.145 | 0.138 |

. | . | Point length (m) . | ||||||
---|---|---|---|---|---|---|---|---|

Elliptical channel diameter (c)
. | Flow depth (D)
. | 19 . | 20 . | 25 . | 30 . | 35 . | 40 . | 45 . |

0.799 | 0.748 | 0.600 | 0.524 | 0.476 | 0.441 | 0.415 | ||

| 0.177 | 0.166 | 0.133 | 0.116 | 0.106 | 0.098 | 0.092 | |

0.811 | 0.758 | 0.607 | 0.530 | 0.478 | 0.447 | 0.428 | ||

| 0.193 | 0.181 | 0.144 | 0.126 | 0.113 | 0.106 | 0.102 | |

0.856 | 0.802 | 0.643 | 0.561 | 0.511 | 0.473 | 0.445 | ||

| 0.204 | 0.191 | 0.153 | 0.133 | 0.121 | 0.112 | 0.106 | |

0.851 | 0.796 | 0.637 | 0.556 | 0.502 | 0.469 | 0.449 | ||

| 0.212 | 0.199 | 0.159 | 0.139 | 0.125 | 0.117 | 0.112 | |

0.945 | 0.885 | 0.707 | 0.618 | 0.558 | 0.522 | 0.499 | ||

| 0.262 | 0.245 | 0.196 | 0.171 | 0.155 | 0.145 | 0.138 |

Also, for the rectangular steep channels in which the ratio of flow depth to channel width is less than 0.16, the formation of roll waves is inevitable (Bazargan & Aghebatie 2015).

By reducing the longitudinal slope from 20 to 17% and decreasing the diameter of the elliptical channel from 4.5 to 3.6 m, it was observed that the hydraulic depth along the channel is increased. This condition induces the turbulence on flow later than the previous case, by decreasing the longitudinal slope and decreasing the width in the rectangular steep open channel. Hence, the elliptical channels are more effective, because they induce less turbulence. As the chutes have steep longitudinal slopes, so the flow is always supercritical and this causes instability in the flow surface, which is a prerequisite for the roll wave formation. Therefore, if the criteria indicate that roll waves will occur, we can replace the rectangular steep channels section with an elliptical section.

## CONCLUSIONS

*k*-

*ɛ*turbulence model, the proposed criteria that can be used as a design condition for chutes or steep open channels. In all the models with elliptical steep open channels with flow depth to channel base ratios of less than 0.14, the formation of roll waves is inevitable. The derived results indicate that for the sections of the steep open channels structure if the angle of the wave forehead with the level vertical is less than 35°, we must expect the formation of the rolling waves.

Results indicated that among the shapes of the steep open channels section proposed to reduce the roll wave formation probability, the elliptical channel is more effective. If the criteria indicate that roll waves will occur, the shape of the section should be modified to reduce the probability of waves being generated. The rectangular steep open channels section should be replaced with an elliptical section.